Problem 23 - Algebra 2 and Trigonometry

Question

In \(15-26,\) write each logarithmic equation in exponential form. $$ -2=\log _{5} 0.04 $$

Step-by-Step Solution

Answer

The exponential form is \(5^{-2} = 0.04\).

Step 1: Understand the Logarithmic Form

The logarithmic equation given is \(-2 = \log_{5} 0.04\). This means that \(5\) raised to the power of \(-2\) gives \(0.04\). In general, \(\log_b a = c\) can be rewritten as \(b^c = a\).

Step 2: Convert to Exponential Form

To convert \(-2 = \log_{5} 0.04\) to its exponential form, use the relationship that \(b^c = a\). Here, \(b = 5\), \(c = -2\), and \(a = 0.04\). So the exponential form is \(5^{-2} = 0.04\).

Key Concepts

Logarithmic Equation

A logarithmic equation involves an expression with a logarithm, and it typically relates three components: a base, a result, and an exponent. The general form is represented as \(\log_b a = c\). In this format, \(b\) represents the base of the logarithm, \(a\) is the value you're taking the logarithm of, and \(c\) is the result or exponent. For example, in the equation \(-2 = \log_{5} 0.04\), \(5\) is the base, \(0.04\) is the result of the base raised to some power, and \(-2\) is the power itself.

• \(b\) is always a positive real number and not equal to 1.
• \(a\) is a positive number because you can’t take the logarithm of zero or a negative number in real numbers.
• \(c\) can be any real number, positive or negative.

Understanding how these elements interact helps in transforming the logarithmic equation into an exponential equation.

Logarithmic to Exponential Conversion

Converting a logarithmic equation to exponential form is a crucial skill in mastering logarithms. This conversion is straightforward once you understand the relationship between the two. You start with a logarithmic equation: \(\log_b a = c\). To convert this into an exponential form, you rewrite it as \(b^c = a\).

For example, the given logarithmic equation is \(-2 = \log_{5} 0.04\). To convert, you recognize:

• \(b = 5\)
• \(c = -2\)
• \(a = 0.04\)

Therefore, the exponential format becomes \(5^{-2} = 0.04\). This shows that multiplying the base \(5\) by itself \(-2\) times results in \(0.04\), effectively reversing the process by which we started.

This conversion process is helpful because it often simplifies solving exponential growth or decay problems.

Exponentiation

Exponentiation is a fundamental mathematical operation involving numbers. It refers to multiplying a number by itself a certain number of times, as indicated by an exponent. When we write \(b^c\), \(b\) is called the base and \(c\) is the exponent. When the exponent is negative, as in \(5^{-2}\), this indicates reciprocal or division.

Using this concept, you calculate \(5^{-2}\) as follows:

• The negative exponent indicates that we take the reciprocal of the base raised to the positive exponent.
• \(5^2 = 25\), hence \(5^{-2} = \frac{1}{25}\).
• Therefore, \(5^{-2} = 0.04\) holds true, as the division yields \(0.04\).

Understanding exponentiation, especially with negative exponents, is essential in mathematics, as it demonstrates how numbers can be depicted in various forms, aiding in simplifying complex equations and solving real-world problems.